#6380649

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,21,17,20,24,18)(15,19,22,27,23,26,28,25);(5,7,9,8)(6,10,11,12)(14,18)(15,22)(16,24)(20,21)(23,28)(25,27);(5,8,9,7)(6,10,11,12);(13,15,16,22,17,23,24,28)(14,19,21,27,20,26,18,25);(14,20)(18,21)(19,26)(25,27)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287845);